Question about the Smallest Grammar problem.

Question

Is the problem to prove whether or not there exists an algorithm with running time polynomial in the length of the input string $|s|$, or polynomial both in $|s|$ and the size of the alphabet $|A|$ ? The papers I'm looking at assume that you know which one they mean.

Edit: Paper

Answer 1

The very first sentence of the paper defines what problem they are considering:

This paper addresses the smallest grammar problem; namely, what is the smallest context-free grammar that generates exactly one given string?